% Unrefereed preprint. No submission or deposit has been made.
% Copyright 2026 Ricardo Pinho. Licensed under CC BY 4.0.
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\title[A certificate for the Schur--Siegel--Smyth constant]{An explicit upper-bound certificate for the Schur--Siegel--Smyth trace constant}
\author{Ricardo Pinho}
\address{Independent researcher}
\thanks{ORCID: \href{https://orcid.org/0009-0006-2744-2599}{0009-0006-2744-2599}. This note is licensed under \href{https://creativecommons.org/licenses/by/4.0/}{CC BY 4.0}.}
\date{19 October 2026; unrefereed preprint}
\keywords{Schur--Siegel--Smyth trace problem, totally positive algebraic integers, logarithmic potential, computer-assisted proof}
\begin{document}
\begin{abstract}
We document an explicit certificate proving $C_{86}<1.8117251$
for the smallest limiting trace-to-degree ratio of totally positive
algebraic integers. The construction uses the primal problem of Orloski
and Talebizadeh Sardari, with their four-polynomial set enlarged to eleven.
A discretised solution is rounded to a rational step density, then
normalised to a probability measure. Exact rational calculations and
rigorous ball-arithmetic inequalities verify Smith's sufficient criterion
for an arithmetic measure. An independently written agent checker accepts
the certificate and rejects four corrupted controls. This note records
the proof, data provenance, reproduction results and limitations of the
previously released certificate; it makes no optimality claim.
\end{abstract}
\maketitle

\section{Statement}
An algebraic integer $\alpha$ of degree $d$ is \emph{totally positive} if
all its conjugates $\alpha_1,\ldots,\alpha_d$ are strictly positive real
numbers. Write $\overline{\tr}(\alpha)=d^{-1}\sum_j\alpha_j$ and define
\begin{equation}\label{eq:constant}
 C_{86}=\sup\{t\in\RR:
   \#\{\alpha\text{ totally positive}:\overline{\tr}(\alpha)<t\}<\infty\}.
\end{equation}
This is the constant denoted $\lambda_{\mathrm{SSS}}$ in the literature
and constant 86a in Tao and contributors' table~\cite{table}.
It concerns a limiting ratio, allowing finitely many exceptional integers.

\begin{theorem}\label{thm:bound}
The Schur--Siegel--Smyth trace constant satisfies
\[
 C_{86}\le \frac{144938007612277893}{79999999999994000}
 =1.811725095153609\ldots <1.8117251.
\]
\end{theorem}

The exact rational number, rather than its decimal expansion, is the
bound proved here. The certificate was already released in~\cite{certificate}
and incorporated into the table through PR~227, merged on 18 October
2026~\cite{pr227}. This paper supplies its mathematical and computational
account. Table inclusion is not journal peer review.

\section{Background}
The classical Schur--Siegel--Smyth problem asked whether the smallest
limiting ratio was $2$. Schur and Siegel established successively stronger
lower bounds, and Smyth developed the auxiliary-polynomial method;
Smith~\cite{smith}, whose introduction reviews this history, disproved
the conjectured value $2$ by proving $C_{86}<1.89831$.
His converse to Fekete's theorem characterises which probability measures
on suitable real compact sets occur as limits of conjugate distributions.

We use the positive logarithmic potential convention
\[
 U_\mu(z)=\int\log|z-y|\,d\mu(y).
\]
Smith uses its negative. A probability measure $\nu$ is called
\emph{arithmetic} here if $\int\log|Q|\,d\nu\ge0$ for every nonzero
$Q\in\ZZ[x]$. Smith's Theorem~1.5 in arXiv v2 states that, on a compact
real set $\Sigma$ with at most countably many components and logarithmic
capacity greater than $1$, this condition is equivalent to being the
weak limit of root-counting probability measures of infinitely many
distinct monic irreducible integer polynomials with roots in $\Sigma$.

Orloski and Talebizadeh Sardari~\cite{os} give a finite-polynomial primal
problem that produces such measures. Their four-polynomial choice
\begin{equation}\label{eq:A4}
 A_4=\{x,1-x,x^2-3x+1,x^3-5x^2+6x-1\}
\end{equation}
gives $C_{86}\le1.8216$. In the checked v2 full text this is
Corollary~1.12: $|\Lambda_{A_4}-1.8215998|\le10^{-7}$.
The present construction uses their framework and seven further
small-trace integer polynomials; it introduces no new converse theorem.

Serre's obstruction, discussed in~\cite[\S1.2]{os} and
\cite[\S5.2]{smith}, puts the ceiling $1.898302\ldots$ on the classical
auxiliary-function lower-bound method. Smith's theorem turns the
associated measure into an upper bound. This ceiling on lower-bound
certificates does not prevent smaller upper bounds obtained from other
arithmetic measures.

In the other direction, Orloski, Talebizadeh Sardari and Smith~\cite{oss}
prove the published lower bound $1.80203$; their article appeared online
in 2024 and in volume~94 (2025) of \emph{Mathematics of Computation}.
Niedbala Giraudin's September 2026 preprint~\cite{giraudin} reports
$1.80220$. We checked these source statements, without independently
reverifying their certificates. Both lower values are below the upper
bound in Theorem~\ref{thm:bound}.

\section{The construction and its implication}
For a finite set $A\subset\ZZ[x]$, the primal problem
of~\cite[\S1.3.1]{os} minimises $\int x\,d\mu$ over nonnegative,
compactly supported measures on $[0,\infty)$ and nonnegative weights $b_Q$,
subject to
\begin{equation}\label{eq:primal}
 U_\mu(z)\ge\sum_{Q\in A}b_Q\log|Q(z)|\quad(z\in\CC),\qquad
 \int\log|Q|\,d\mu\ge0\ (Q\in A),\qquad \mu(\RR)\ge1.
\end{equation}
The rounded measure below has mass slightly less than $1$. Thus it is
the pair $(\mu/m,(b_Q/m)_{Q\in A})$, where $m=\mu(\RR)$, that is
feasible. All potential and logarithmic-integral inequalities are
homogeneous under this simultaneous scaling.

The data are in \texttt{cert\_n1200\_A11.json}~\cite{certificate}.
The grid is $e_i=i/200$, $0\le i\le1200$, and
\begin{equation}\label{eq:density}
 d\mu(x)=\sum_{i=0}^{1199}d_i\,\boldsymbol{1}_{[e_i,e_{i+1}]}(x)\,dx.
\end{equation}
All $d_i$ and $b_Q$ are nonnegative rationals with denominator dividing
$10^{12}$. There are exactly $629$ positive-density cells; their union
has $25$ connected components and lies in $[1/20,1037/200]$.
The complete density array is specified by the hashed certificate, rather
than transcribed here. Table~\ref{tab:polys} gives every polynomial and
weight in its order in that file; the decimal weights are exact.

\begin{table}[ht]
\centering\small
\caption{The eleven polynomials and the unnormalised rational weights $b_Q$.
The first four are~\eqref{eq:A4}.}\label{tab:polys}
\begin{tabular}{@{}lr@{}}
\toprule
$Q(x)$ & $b_Q$\\
\midrule
$x$ & $0.038759530583$\\
$1-x$ & $0.079868388122$\\
$x^2-3x+1$ & $0.046317667230$\\
$x^3-5x^2+6x-1$ & $0.053369420893$\\
\midrule
$x^4-7x^3+14x^2-8x+1$ & $0.019753937664$\\
$x^4-7x^3+13x^2-7x+1$ & $0.036545507753$\\
$x^5-9x^4+28x^3-35x^2+15x-1$ & $0.016049367094$\\
$x^5-9x^4+26x^3-29x^2+11x-1$ & $0.009731525491$\\
$x^5-9x^4+27x^3-32x^2+13x-1$ & $0.007106935573$\\
$x^3-6x^2+9x-1$ & $0.018100957401$\\
$x^5-9x^4+27x^3-31x^2+12x-1$ & $0.002893658474$\\
\bottomrule
\end{tabular}
\end{table}

The construction code solves a sampled linear program on this grid,
adds points with deficient potential slack, and uses target slack
$\delta=10^{-4}$. Cell densities are rounded to the nearest multiple of
$10^{-12}$ and weights down to multiples of $10^{-12}$. These search
steps only produce a candidate: feasibility is established afterwards
on entire intervals. Exact summation gives
\begin{align}\label{eq:moments}
 m&=\frac{39999999999997}{40000000000000},&
 T:=\int x\,d\mu&=\frac{144938007612277893}{80000000000000000},\\
 \sum_{Q\in A}b_Q\deg Q&=\frac{6638236823}{8000000000}<m.\nonumber
\end{align}

For clarity, we give the reduction from finite checks to the assertion
about algebraic integers, rather than assuming the candidate is an
optimiser of the primal problem.

\begin{proposition}\label{prop:criterion}
Suppose $\mu$ is a nonzero bounded-density measure on a compact subset
$K\subset(0,\infty)$, and $A$ consists of pairwise nonassociate primitive
irreducible integer polynomials. Suppose $b_Q>0$,
$\sum_Qb_Q\deg Q<m=\mu(\RR)$, $\int\log|Q|\,d\mu\ge0$ for each
$Q\in A$, and
\begin{equation}\label{eq:f}
 f(x):=U_\mu(x)-\sum_{Q\in A}b_Q\log|Q(x)|\ge0\quad(x\in K).
\end{equation}
If $K\subset\Sigma\subset(0,\infty)$ for a real compact interval
$\Sigma$ of capacity greater than $1$, then $C_{86}\le T/m$.
\end{proposition}
\begin{proof}
Off $K$, $U_\mu$ is harmonic and each $-b_Q\log|Q|$ is superharmonic,
with value $+\infty$ at its roots. The bounded density makes $U_\mu$
continuous on $\CC$. At infinity,
\[
 f(z)=\left(m-\sum_Qb_Q\deg Q\right)\log|z|
       -\sum_Qb_Q\log|\operatorname{lc}(Q)|+o(1)\longrightarrow+\infty.
\]
The minimum principle on a sufficiently large disk with $K$ removed
therefore extends~\eqref{eq:f} to all $z\in\CC$.

Put $\nu=\mu/m$ and $a_Q=b_Q/m$. For any primitive irreducible
$P\in\ZZ[x]$ not associated to an element of $A$, write
$P(x)=c\prod_{j=1}^n(x-\alpha_j)$. The nonzero integer resultants give
\begin{align*}
 \int\log|P|\,d\nu
 &=\log|c|+\sum_jU_\nu(\alpha_j)\\
 &\ge\left(1-\sum_Qa_Q\deg Q\right)\log|c|
       +\sum_Qa_Q\log|\Res(P,Q)|\ge0.
\end{align*}
For the elements of $A$ the direct integral constraints give this
inequality; integer constants and products then give it for every
nonzero integer polynomial. This is Smith's sufficiency argument
\cite[Proposition~5.15, v2]{smith}.

Apply Smith's Theorem~1.5 to $\nu$ on $\Sigma$. Its distinct monic
irreducible polynomials define distinct totally positive algebraic
integers. Weak convergence on the fixed compact interval preserves the
mean, so their trace-to-degree ratios tend to $\int x\,d\nu=T/m$.
Definition~\eqref{eq:constant} gives the assertion.
\end{proof}

For our data take $K=\supp\mu$ and
$\Sigma=[1/20,1037/200]$. Its capacity is its length divided by four,
namely $1027/800>1$. The verification below establishes the remaining
hypotheses. Equation~\eqref{eq:moments} then proves
Theorem~\ref{thm:bound}.

\section{Verification}
The transcendental quantities are evaluated from closed forms, without
numerical quadrature. Put $G(u)=u\log|u|-u$, with $G(0)=0$. A cell's
potential is
\[
 H_i(x)=G(x-e_i)-G(x-e_{i+1}).
\]
It decreases to its cell midpoint and increases thereafter. On any real
interval $[s,t]$, its minimum is attained at an endpoint or at that
midpoint, if present. For a real root $r$, $\log|x-r|$ is bounded above
by the logarithm of the larger endpoint distance. Consequently,
\begin{equation}\label{eq:interval}
 f(x)\ge\sum_i d_i\min_{[s,t]}H_i
   -\sum_Qb_Q\sum_{Q(r)=0}\log\max\{|s-r|,|t-r|\}
 \quad(s\le x\le t),
\end{equation}
since all leading coefficients in Table~\ref{tab:polys} have modulus $1$.
Adaptive subdivision certifies intervals until their lower bounds
are positive. This proves a continuum inequality, including cell edges.

The original \texttt{verify\_cert.py} rederives positivity, mass, moments
and the condition at infinity using exact fractions, computes the eleven
logarithmic integrals with 50-digit \texttt{mpmath}, and checks
\eqref{eq:interval} in floating point with an allowance of $10^{-8}$.
It is a diagnostic cross-check, not by itself a rigorous transcendental
verifier. The companion \texttt{verify\_arb.py} uses floating point only
to choose a partition, then recomputes its lower bounds in 128-bit Arb
ball arithmetic. Endpoint balls of radius $10^{-15}$ cover the rounded
dyadic subdivisions of the rational grid cells. The fresh run returned
the following excerpts:
\begin{quote}\small\ttfamily
|A|=11 cells=629 (2) min=0.0001\\
(3) 155719 subintervals, depth 10\\
exact = 144938007612277893/79999999999994000\\
arb check: 78174 subintervals, failures 0,\\
min LB lower endpoint 1.133e-08\\
STEP3\_RIGOROUS\_OK
\end{quote}
The first count includes unsuccessful parent intervals; the Arb count
is the accepted partition. The rigorous direct-integral and algebraic
checks are supplied by the independent checker described next.

An independent AI-agent referee wrote \texttt{referee\_check.py} from
scratch, without importing the construction code. Its released version
in~\cite{certificate} checks increasing edges, nonnegative densities and
weights, support away from zero, exact mass and trace, the exact claimed
bound, irreducibility, pairwise coprimality and real-root counts. It
isolates roots by exact rational intervals of width at most $10^{-35}$
and encloses the eleven direct integrals using Arb. All lie in
$[0.0000999999986,0.0001000000008]$, hence are strictly positive.

For the potential inequality it uses a different enclosure: rewrite
$U_\mu$ as a sum of $G$ terms at density jumps, bound nearby singular
terms directly, and use a midpoint value and an interval derivative for
the remaining terms. The fresh run reports
\begin{quote}\small\ttfamily
E4 ok\\
E5: cells=629, subintervals\_proved=6218,\\
min\_certified\_lower\_bound=7.359790864206603e-08,\\
max\_depth=5, n\_failures=0\\
ALL\_OK True
\end{quote}
The four corrupted-certificate controls were also rerun. Multiplying
densities by $0.98$ produces a certified negative value of $f$;
multiplying weights by $1.3$ fails the condition at infinity; adding
$1/10$ to the first positive cell density fails a direct logarithmic
integral and produces negative potential differences; subtracting
$10^{-12}$ from the claimed bound fails exact equality with $T/m$.
All four were rejected. This agent audit is not external peer review.

\newpage
\section{Reproducibility}
The public repository is \url{https://github.com/rpinho/c86-trace-certificate},
at commit
\begin{quote}\small\ttfamily
116a58ff99681226ce6299d4a02eeff1e7d3c5c7.
\end{quote}
The SHA-256 of its \texttt{cert\_n1200\_A11.json} is
\begin{quote}\small\ttfamily
722a287b74a9f84f8c6c2f505e7e6bb987ca\\
634710b031287a5b01619bffe53d.
\end{quote}
The line break is typographical. A fresh clone was checked on
19 October 2026 with both original scripts copied unchanged from the
construction archive, and with the public referee script. The original
and public JSON files parse to identical objects; their byte hashes
differ because of formatting. The hash above identifies the public file.
The public repository contains the certificate and referee checker;
the original pair and complete replay logs are retained with this note's
local reproduction report.

From a fresh checkout with its requirements installed, the public
verification command is
\begin{quote}\small\ttfamily
python -I referee\_check.py cert\_n1200\_A11.json 2
\end{quote}
The final argument limits the pool to two workers. The archived original
commands are
\begin{quote}\small\ttfamily
python verify\_cert.py CERT.json\\
python verify\_arb.py CERT.json 2
\end{quote}
Limit BLAS and OpenMP to one thread per process.
Neither the floating-point optimiser nor its search history
is needed to check the released certificate.

\section{Limitations and prior-art check}
This is an upper bound only. We prove neither optimality for the
eleven-polynomial problem nor a matching lower bound for $C_{86}$.
We do not explicitly list a sequence of minimal polynomials achieving
the certified limiting distribution. The mathematical implication uses
Smith's published theorem and the minimum-principle argument above;
the software checks are not a formal proof in a proof assistant.

The live 86a source and open issues/PRs were checked on 19 October 2026.
The source lists $1.8117251$ with an asterisk for a self-archived,
unrefereed certificate, and $1.8216$ as the best unasterisked upper bound;
the lower-bound update to $1.80220$ remains open as PR~223.
The checked arXiv records and full texts include~\cite{smith,os,oss,giraudin};
recent arXiv and public-code searches did not locate a smaller upper
bound than this certificate. Search coverage does not prove priority.

Orloski's seminar abstract at Hebrew University, dated 2 January 2025,
announces $1.81618$~\cite{seminar}.
Penn State's official Ph.D. list confirms his 2025 dissertation,
\emph{Distributions of Conjugates of Algebraic Integers and the
Schur--Siegel--Smyth Trace Problem}~\cite{thesisstatus}.
Its library search record is indexed as open access, but direct access
was blocked during this audit; the thesis's numerical contents remain
unchecked. Thus we do not assert exhaustive priority over the thesis.
This limitation does not affect the bound proved by the certificate.

\section*{AI assistance and author review}
Analysis and proof drafting by AI agents under my direction; I reviewed the results at summary level.
The construction, software verification and independent agent audit
also used AI assistance. Independence here means separate code and
reasoning, not an unaffiliated human referee.

\newpage
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